In the general problem, a desired target is buried in both interference and noise. A transmit signal excites both the desired target and the interference simultaneously. The interference and/or interferences can be foliage returns in the form of clutter for radar, scattered returns of the transmit signal from a sea-bottom and different ocean-layers in the case of sonar, or multipath returns in a communication scene. In all of these cases, like the target return, the interference returns are also transmit signal dependent, and hence it puts conflicting demands on the receiver. In general, the receiver input is comprised of target returns, interferences and the ever present noise. The goal of the receiver is to enhance the target returns and simultaneously suppress both the interference and noise signals. In a detection environment, a decision regarding the presence or absence of a target is made at some specified instant t=t0 using output data from a receiver, and hence to maximize detection, the Signal power to average Interference plus Noise Ratio (SINR) at the receiver output can be used as an optimization goal. This scheme is illustrated in FIG. 1.
The transmitter output bandwidth can be controlled using a known transmitter output filter having a transfer function P1(ω) (see FIG. 2A). A similar filter with transform characteristics P2(ω) can be used at a receiver input 22a shown in FIG. 1, to control the processing bandwidth as well.
The transmit waveform set f(t) at an output 10a of FIG. 1, can have spatial and temporal components to it each designated for a specific goal. The simplest situation is that shown in FIG. 2A where a finite duration waveform f(t) of energy E is to be designed. Thus
                                          ∫            0                          T              o                                ⁢                                                                                      f                  ⁡                                      (                    t                    )                                                                              2                        ⁢                          ⅆ              t                                      =                  E          .                                    (        1        )            
Usually, transmitter output filter 12 characteristics P1(ω), such as shown in FIG. 2B, are known and for design purposes, it is best to incorporate the transmitter output filter 12 and the receiver input filter (which may be part of receiver 22) along with the target and clutter spectral characteristics.
Let q(t)Q(ω) represent the target impulse response and its transform. In general q(t) can be any arbitrary waveform. Thus the modified target that accounts for the target output filter has transform P1(ω)Q(ω) etc. In a linear domain setup, the transmit signal f(t) interacts with the target q(t), or target 14 shown in FIG. 1, to generate the output below (referred to in S. U. Pillai, H. S. Oh, D. C. Youla, and J. R. Guerci, “Optimum Transmit-Receiver Design in the Presence of Signal-Dependent Interference and Channel Noise”, IEEE Transactions on Information Theory, Vol. 46, No. 2, pp. 577-584, March 2000 and J. R. Guerci and S. U. Pillai, “Theory and Application of Optimum Transmit-Receive Radar,” IEEE International Radar Conference, Alexandria Va., May 2000, pp. 705-710):
                              s          ⁡                      (            t            )                          =                                            f              ⁡                              (                t                )                                      *                          q              ⁡                              (                t                )                                              =                                    ∫              0                              T                o                                      ⁢                                          f                ⁡                                  (                  τ                  )                                            ⁢                              q                ⁡                                  (                                      t                    -                    τ                                    )                                            ⁢                              ⅆ                τ                                                                        (        2        )            that represents the desired signal.
The interference returns are usually due to the random scattered returns of the transmit signal from the environment, and hence can be modeled as a stochastic signal wc(t) that is excited by the transmit signal f(t). If the environment returns are stationary, then the interference can be represented by its power spectrum Gc(ω). This gives the average interference power to be Gc(ω)|F(ω)|2. Finally let n(t) represent the receiver 22 input noise with power spectral density Gn(ω). Thus the receiver input signal at input 22a equalsr(t)=s(t)+wc(t)*f(t)+n(t),  (3)and the input interference plus noise power spectrum equalsGI(ω)=Gc(ω)|F(ω)|2+Gn(ω).  (4)The received signal is presented to the receiver 22 at input 22a with impulse response h(t). The simplest receiver is of the noncausal type.
With no restrictions on the receiver 22 of FIG. 1, its output signal at output 22b in FIG. 1, and interference noise components are given by
                                                        y              s                        ⁡                          (              t              )                                =                                                    s                ⁡                                  (                  t                  )                                            *                              h                ⁡                                  (                  t                  )                                                      =                                          1                                  2                  ⁢                  π                                            ⁢                                                ∫                                      -                    ∞                                                        +                    ∞                                                  ⁢                                                      S                    ⁡                                          (                      ω                      )                                                        ⁢                                      H                    ⁡                                          (                      ω                      )                                                        ⁢                                      ⅇ                                          jω                      ⁢                                                                                          ⁢                      t                                                        ⁢                                      ⅆ                    ω                                                                                      ⁢                                  ⁢                  and                                    (        5        )                                                      y            n                    ⁡                      (            t            )                          =                              {                                                                                w                    c                                    ⁡                                      (                    t                    )                                                  *                                  f                  ⁡                                      (                    t                    )                                                              +                              n                ⁡                                  (                  t                  )                                                      }                    *                                    h              ⁡                              (                t                )                                      .                                              (        6        )            The output yn(t) represents a second order stationary stochastic process with power spectrum below (referred to in the previous publications and in Athanasius Papoulis, S. Unnikrishna Pillai, Probability, Random Variables and Stochastic Processes, McGraw-Hill Higher Education, New York 2002):Go(ω)=(Gc(ω)|F(ω)|2+Gn(ω)|H(ω)|2  (7)and hence the total output interference plus noise power is given by
                                                                        σ                                  I                  +                  N                                2                            =                                                1                                      2                    ⁢                    π                                                  ⁢                                                      ∫                                          -                      ∞                                                              +                      ∞                                                        ⁢                                                                                    G                        0                                            ⁡                                              (                        ω                        )                                                              ⁢                                          ⅆ                      ω                                                                                                                                              =                                                1                                      2                    ⁢                    π                                                  ⁢                                                      ∫                                          -                      ∞                                                              +                      ∞                                                        ⁢                                                            (                                                                                                                                  G                              C                                                        ⁡                                                          (                              ω                              )                                                                                ⁢                                                                                                                                                  F                                ⁡                                                                  (                                  ω                                  )                                                                                                                                                    2                                                                          +                                                                              G                            n                                                    ⁡                                                      (                            ω                            )                                                                                              )                                        ⁢                                                                                                                    H                          ⁡                                                      (                            ω                            )                                                                                                                      2                                        ⁢                                                                  ⅆ                        ω                                            .                                                                                                                              (        8        )            Referring back to FIG. 1, the signal component ys(t) in equation (5) at the receiver output 22b needs to be maximized at the decision instant to in presence of the above interference and noise. Hence the instantaneous output signal power at t=to is given by the formula below shown in S. U. Pillai, H. S. Oh, D. C. Youla, and J. R. Guerci, “Optimum Transmit-Receiver Design in the Presence of Signal-Dependent Interference and Channel Noise”, IEEE Transactions on Information Theory, Vol. 46, No. 2, pp. 577-584, March 2000, which is incorporated by reference herein:
                              P          O                =                                                                                            y                  s                                ⁡                                  (                                      t                    o                                    )                                                                    2                    =                                                                                                          1                                          2                      ⁢                      π                                                        ⁢                                                            ∫                                              -                        ∞                                                                    +                        ∞                                                              ⁢                                                                  S                        ⁡                                                  (                          ω                          )                                                                    ⁢                                              H                        ⁡                                                  (                          ω                          )                                                                    ⁢                                              ⅇ                                                  jω                          ⁢                                                                                                          ⁢                                                      t                            o                                                                                              ⁢                                              ⅆ                        ω                                                                                                                        2                        .                                              (        9        )            This gives the receiver output SINR at t=to be the following as specified in Guerci et. al., “Theory and Application of Optimum Transmit-Receive Radar”, pp. 705-710; and Pillai et. al., “Optimum Transmit-Receiver Design in the Presence of Signal-Dependent Interference and Channel Noise”, incorporated herein by reference:
                    SINR        =                                            P              o                                      σ                              I                +                N                            2                                =                                                                                                                                  1                                              2                        ⁢                        π                                                              ⁢                                                                  ∫                                                  -                          ∞                                                                          +                          ∞                                                                    ⁢                                                                        S                          ⁡                                                      (                            ω                            )                                                                          ⁢                                                  H                          ⁡                                                      (                            ω                            )                                                                          ⁢                                                  ⅇ                                                      jω                            ⁢                                                                                                                  ⁢                                                          t                              o                                                                                                      ⁢                                                  ⅆ                          ω                                                                                                                                      2                                                              1                                      2                    ⁢                    π                                                  ⁢                                                      ∫                                          -                      ∞                                                              +                      ∞                                                        ⁢                                                                                    G                        I                                            ⁡                                              (                        ω                        )                                                              ⁢                                                                                                                    H                          ⁡                                                      (                            ω                            )                                                                                                                      2                                        ⁢                                          ⅆ                      ω                                                                                            .                                              (        10        )            We can apply Cauchy-Schwarz inequality in equation (10) to eliminate H(ω). This gives
                                                        SINR              ≤                            ⁢                                                1                                      2                    ⁢                    π                                                  ⁢                                                      ∫                                          -                      ∞                                                              +                      ∞                                                        ⁢                                                                                                                                                                    S                            ⁡                                                          (                              ω                              )                                                                                                                                2                                                                                              G                          I                                                ⁡                                                  (                          ω                          )                                                                                      ⁢                                          ⅆ                      ω                                                                                                                                              =                            ⁢                                                1                                      2                    ⁢                    π                                                  ⁢                                                      ∫                                          -                      ∞                                                              +                      ∞                                                        ⁢                                                                                                                                                                                                    Q                              ⁡                                                              (                                ω                                )                                                                                                                                          2                                                ⁢                                                                                                                                        F                              ⁡                                                              (                                ω                                )                                                                                                                                          2                                                                                                                                                                                G                              c                                                        ⁡                                                          (                              ω                              )                                                                                ⁢                                                                                                                                                  F                                ⁡                                                                  (                                  ω                                  )                                                                                                                                                    2                                                                          +                                                                              G                            n                                                    ⁡                                                      (                            ω                            )                                                                                                                ⁢                                          ⅆ                      ω                                                                                                                                              =                            ⁢                                                SINR                  max                                .                                                                        (        11        )            Thus the maximum obtainable SINR is given by equation (11), and this is achieved if and only if the following equation referred to in previous prior art publications, is true:
                                                                                          H                  opt                                ⁡                                  (                  ω                  )                                            =                                                                    S                    *                                          (                      ω                      )                                                                                                                                                    G                          c                                                ⁡                                                  (                          ω                          )                                                                    ⁢                                                                                                                              F                            ⁡                                                          (                              ω                              )                                                                                                                                2                                                              +                                                                  G                        n                                            ⁡                                              (                        ω                        )                                                                                            ⁢                                  ⅇ                                                            -                      jω                                        ⁢                                                                                  ⁢                                          t                      o                                                                                                                                              =                                                                    Q                    *                                          (                      ω                      )                                        ⁢                    F                    *                                          (                      ω                      )                                                                                                                                                    G                          c                                                ⁡                                                  (                          ω                          )                                                                    ⁢                                                                                                                              F                            ⁡                                                          (                              ω                              )                                                                                                                                2                                                              +                                                                  G                        n                                            ⁡                                              (                        ω                        )                                                                                            ⁢                                                      ⅇ                                                                  -                        jω                                            ⁢                                                                                          ⁢                                              t                        o                                                                              .                                                                                        (        12        )            In (12), the phase shift e−jωto can be retained to approximate causality for the receiver waveform. Interestingly even with a point target (Q(ω)≡1), flat noise (Gn(ω)=σn2), and flat clutter (Gc(ω)=σc2), the optimum receiver is not conjugate-matched to the transmit signal, since in that case from equation (12) we have the following formula given by Pillai et. al., “Optimum Transmit-Receiver Design in the Presence of Signal-Dependent Interference and Channel Noise”, incorporated herein by reference, Papoulis, “Probability, Random Variables and Stochastic Processes”, and H. L. Van Trees, Detection, Estimation, and Modulation Theory, Part 1, New York: John Wiley and Sons, 1968, incorporated by reference:
                                          H            opt                    ⁡                      (            ω            )                          =                                                            F                *                                  (                  ω                  )                                                                                                  σ                    c                    2                                    ⁢                                                                                                          F                        ⁡                                                  (                          ω                          )                                                                                                            2                                                  +                                  σ                  n                  2                                                      ⁢                          ⅇ                                                -                  jω                                ⁢                                                                  ⁢                                  t                  o                                                              ≠                      F            *                          (              ω              )                        ⁢                                          ⅇ                                                      -                    jω                                    ⁢                                                                          ⁢                                      t                    o                                                              .                                                          (        13        )            Prior Art Transmitter Waveform Design
When the receiver design satisfies equation (12), the output SINR is given by the right side of the equation (11), where the free parameter |F(ω)|2 can be chosen to further maximize the output SINR, subject to the transmit energy constraint in (1). Thus the transmit signal design reduces to the following optimization problem:
Maximize
                                          SINR            max                    =                                    1                              2                ⁢                π                                      ⁢                                          ∫                                  -                  ∞                                                  +                  ∞                                            ⁢                                                                                                                                                                    Q                          ⁡                                                      (                            ω                            )                                                                                                                      2                                        ⁢                                                                                                                    F                          ⁡                                                      (                            ω                            )                                                                                                                      2                                                                                                                                                    G                          c                                                ⁡                                                  (                          ω                          )                                                                    ⁢                                                                                                                              F                            ⁡                                                          (                              ω                              )                                                                                                                                2                                                              +                                                                  G                        n                                            ⁡                                              (                        ω                        )                                                                                            ⁢                                  ⅆ                  ω                                                                    ,                            (        14        )            subject to the energy constraint
                                          ∫            0                          T              o                                ⁢                                                                                      f                  ⁡                                      (                    t                    )                                                                              2                        ⁢                          ⅆ              t                                      =                                            1                              2                ⁢                π                                      ⁢                                          ∫                                  -                  ∞                                                  +                  ∞                                            ⁢                                                                                                              F                      ⁡                                              (                        ω                        )                                                                                                  2                                ⁢                                  ⅆ                  ω                                                              =                      E            .                                              (        15        )            To solve this new constrained optimization problem, combine (14)-(15) to define the modified Lagrange optimization function (referred to in T. Kooij, “Optimum Signal in Noise and Reverberation”, Proceeding of the NATO Advanced Study Institute on Signal Processing with Emphasis on Underwater Acoustics, Vol. I, Enschede, The Netherlands, 1968.)
                              Λ          =                                    ∫                              -                ∞                                            +                ∞                                      ⁢                                          {                                                                                                                                                                                    Q                            ⁡                                                          (                              ω                              )                                                                                                                                2                                            ⁢                                                                        y                          2                                                ⁡                                                  (                          ω                          )                                                                                                                                                                                          G                            c                                                    ⁡                                                      (                            ω                            )                                                                          ⁢                                                                              y                            2                                                    ⁡                                                      (                            ω                            )                                                                                              +                                                                        G                          n                                                ⁡                                                  (                          ω                          )                                                                                                      -                                                            1                                              λ                        2                                                              ⁢                                                                  y                        2                                            ⁡                                              (                        ω                        )                                                                                            }                            ⁢                              ⅆ                ω                                                    ⁢                                  ⁢                  where                                    (        16        )                                          y          ⁡                      (            ω            )                          =                                        F            ⁡                          (              ω              )                                                                    (        17        )            is the free design parameter. From (16) (17),
            ∂      Λ              ∂      y        =      0    ⁢                  ⁢    gives  
                                          ∂                          Λ              ⁡                              (                ω                )                                                          ∂            y                          =                              2            ⁢                          y              ⁡                              (                ω                )                                      ⁢                          {                                                                                                                  G                        n                                            ⁡                                              (                        ω                        )                                                              ⁢                                                                                                                    Q                          ⁡                                                      (                            ω                            )                                                                                                                      2                                                                                                  {                                                                                                                                  G                              c                                                        ⁡                                                          (                              ω                              )                                                                                ⁢                                                      y                            2                                                                          -                                                  (                          ω                          )                                                -                                                                              G                            n                                                    ⁡                                                      (                            ω                            )                                                                                              }                                        2                                                  -                                  1                                      λ                    2                                                              }                                =          0.                                    (        18        )            where Λ(ω) represents the quantity within the integral in (16). From (18), either
                              y          ⁡                      (            ω            )                          =        0                            (        19        )                        or                                                                                                                                                G                    n                                    ⁡                                      (                    ω                    )                                                  ⁢                                                                                                Q                      ⁡                                              (                        ω                        )                                                                                                  2                                                                              {                                                                                                              G                          c                                                ⁡                                                  (                          ω                          )                                                                    ⁢                                                                        y                          2                                                ⁡                                                  (                          ω                          )                                                                                      +                                                                  G                        n                                            ⁡                                              (                        ω                        )                                                                              }                                2                                      -                          1                              λ                2                                              =          0                ,                            (        20        )            which gives
                                          y            2                    ⁡                      (            ω            )                          -                                                                              G                  n                                ⁡                                  (                  ω                  )                                                      ⁢                          (                                                λ                  ⁢                                                                                Q                      ⁡                                              (                        ω                        )                                                                                                                -                                                                            G                      n                                        ⁡                                          (                      ω                      )                                                                                  )                                                          G              c                        ⁡                          (              ω              )                                                          (        21        )            provided y2(ω)>0. See T. Kooij cited above incorporated by reference herein.